Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Please show all work and answer the following in Matlab code !!!
For part c, clearly state the root with linear convergence so that part d may be executed.
Please allow the codes to be copied directly into Matlab
This is a part of an ungraded test review so please help!!
![Given the polynomial function:
\[ f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4. \]
### Tasks:
**(b)** Apply Newton's Method to find all five roots within the interval with an accuracy of eight decimal places.
**(c)** For each root, use the value obtained in part (b) to approximate its exact value \( r \). Define the error as \( e_i = |x_i - r| \), which represents the absolute error at step \( i \) in part (b). Plot the curve \( (e_i, e_{i+1}) \). Utilize a logarithmic scale for both the x-axis and y-axis. Analyze the convergence behavior based on the slope of these curves.
**(d)** For the root with linear convergence observed in part (c), repeat the analysis using the secant method.](https://content.bartleby.com/qna-images/question/db82eee3-c597-43b7-b98f-cffc18fbca72/7513a356-4d01-4b3d-abcc-2473a000ebb4/zmty12e_processed.png)
Transcribed Image Text:Given the polynomial function:
\[ f(x) = 54x^6 + 45x^5 - 102x^4 - 69x^3 + 35x^2 + 16x - 4. \]
### Tasks:
**(b)** Apply Newton's Method to find all five roots within the interval with an accuracy of eight decimal places.
**(c)** For each root, use the value obtained in part (b) to approximate its exact value \( r \). Define the error as \( e_i = |x_i - r| \), which represents the absolute error at step \( i \) in part (b). Plot the curve \( (e_i, e_{i+1}) \). Utilize a logarithmic scale for both the x-axis and y-axis. Analyze the convergence behavior based on the slope of these curves.
**(d)** For the root with linear convergence observed in part (c), repeat the analysis using the secant method.
Expert Solution
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Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step 1: Analyze the Graph of the function
VIEW Step 2: Part (b) - Approximate all five roots using Newton's method.
VIEW Step 3: Part (c) Study the convergence of the root approximations
VIEW Step 4: (contd.) Part (c) Study the convergence of the root approximations
VIEW Step 5: (contd.) Part (c) Study the convergence of the root approximations
VIEW Step 6: (contd.) part (c) Analyze the graphs
VIEW Step 7: (contd.) part (c) Approximate the roots using secant method
VIEW Solution
VIEW 
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